Ordered Pairs And Cartesian Products

Ordered Pairs & Cartesian Product | Grade 9 Additional Maths | NMA

Part I — Ordered Pairs

Observation

You have a movie ticket. It contains two pieces of information:

Your row number is 3
Your seat number is 10

We write this information inside parentheses, separated by a comma:

(3,\ 10)

This is a pair of information. If you go to the 10th row looking for seat 3, you end up in someone else's seat — so the order matters!

Conclusion

A pair of information written inside parentheses, separated by a comma, following a specific order is called an Ordered Pair.

Definition

An ordered pair is a pair of numbers, objects, elements or data written in a specific order (rule), usually inside parentheses and separated by a comma, like \((x,\ y)\).

The first element of an ordered pair is called the ANTECEDENT.
The second element of an ordered pair is called the CONSEQUENT.

Equality of Ordered Pairs

Two ordered pairs \((a,\ b)\) and \((c,\ d)\) are said to be equal if and only if:

\(a = c\) — antecedent equals antecedent
\(b = d\) — consequent equals consequent

Example:

Are \(\left(\sqrt{25},\ \sqrt{36}\right)\) and \((5,\ 6)\) equal?

1st Ordered Pair

\(\left(\sqrt{25},\ \sqrt{36}\right)\)

Antecedent:\(\sqrt{25} = 5\)

Consequent:\(\sqrt{36} = 6\)

2nd Ordered Pair

\((5,\ 6)\)

Antecedent:\(5\)

Consequent:\(6\)

✅ Antecedents are equal (5 = 5) & Consequents are equal (6 = 6)
∴ \(\left(\sqrt{25},\ \sqrt{36}\right) = (5,\ 6)\)

Part II — Cartesian Product

Activity

Question: If you have 2 shirts and 3 pants, how many different outfits can you make?

Outfit 1Shirt 1 + Pant A

Outfit 2Shirt 2 + Pant A

Outfit 3Shirt 1 + Pant B

Outfit 4Shirt 2 + Pant B

Outfit 5Shirt 1 + Pant C

Outfit 6Shirt 2 + Pant C

Total outfits: 6

Observation

We have two well-defined, distinct collections:

A = \{S_1,\ S_2\} \qquad B = \{P_A,\ P_B,\ P_C\}

Pairing one item from the first group with one from the second gives a collection of ordered pairs:

(S_1,P_A),\ (S_2,P_A),\ (S_1,P_B),\ (S_2,P_B),\ (S_1,P_C),\ (S_2,P_C)

All pairs are ordered pairs because the order matters (shirt first, pant second).

Conclusion

The collection of all ordered pairs formed by taking one element from each of two sets is known as the Cartesian Product.

Definition

Let \(A\) and \(B\) be any two non-empty sets. The Cartesian Product of \(A\) and \(B\), written as \(A \times B\) (read as "A cross B"), is the collection of all ordered pairs \((a,\ b)\) such that \(a \in A\) and \(b \in B\).

A \times B = \{(a,b) \mid a \in A,\ b \in B\}

⚠️ Important: Order matters in ordered pairs. \((S_1, P_A) \neq (P_A, S_1)\) — therefore, in general, \(\boldsymbol{A \times B \neq B \times A}\).

Cardinality of Cartesian Product

If \(n(A) = m\) and \(n(B) = n\), then the number of ordered pairs in the Cartesian Product is:

n(A \times B) = n(A) \times n(B) = m \times n

In our outfit example:

n(A) — Shirts

n(B) — Pants

n(A×B) — Outfits

Ways of Representing Cartesian Product

Let \(A = \{1, 2, 3\}\) and \(B = \{a, b\}\). Then \(A \times B\) can be represented in five ways:

A \times B = \{(1,a),\ (1,b),\ (2,a),\ (2,b),\ (3,a),\ (3,b)\}

We list all possible ordered pairs, taking the antecedent from \(A\) and the consequent from \(B\).

\(A \times B\)	\(b = a\)	\(b = b\)
\(a = 1\)	\((1, a)\)	\((1, b)\)
\(a = 2\)	\((2, a)\)	\((2, b)\)
\(a = 3\)	\((3, a)\)	\((3, b)\)

Rows represent elements of \(A\); columns represent elements of \(B\). Each cell gives one ordered pair.

Every element of \(A\) is mapped to every element of \(B\).

Each branch of the tree traces one ordered pair.

Each ordered pair \((a, b)\) is plotted as a point on the coordinate plane. Elements of \(A\) go on the horizontal axis; elements of \(B\) on the vertical axis.

Quick Reference

Concept	Symbol / Expression	Key Point
Ordered Pair	\((x, y)\)	Order matters: \((x,y) \neq (y,x)\) if \(x\neq y\)
Antecedent	1st element	Left side of the comma
Consequent	2nd element	Right side of the comma
Equality	\((a,b)=(c,d)\)	\(a=c\) and \(b=d\)
Cartesian Product	\(A \times B\)	All ordered pairs \((a,b)\) with \(a\!\in\!A,\ b\!\in\!B\)
Cardinality	\(n(A\times B)\)	\(= n(A)\times n(B)\)